$\newcommand{\A}{\mathscr{A}}\newcommand{\I}{\mathscr{I}}\newcommand{\psh}{\mathsf{Psh}}\newcommand{\Set}{\mathsf{Set}}\newcommand{\op}{^\mathsf{op}}\newcommand{\ev}{\mathsf{ev}}$I am studying category theory from Tom Leinster's Basic Category Theory. In it, since it is a "basic" text, size issues are not discussed in detail, but briefly covered. However in chapter $6$ he has a strange, unexplained insistence on smallness. An important example ($\psh_\A=[\A\op,\Set]$ the category of presheaves):
$(\ast)$ Let $\A$ be a small category. Then $\psh_\A$ has all limits and colimits, and they are preserved (and jointly reflected) by all evaluation functors $\ev_A:\psh_\A\to\Set,\,X\mapsto X(A)$, $A\in\A$.
This is a corollary of a more general result:
Let $\A,\mathbf{I}$ be small categories and $\I$ a locally small category. Suppose for some diagram $D:I\to[\A,\I]$ that for every $A\in\A$, the composite diagram $\ev_A\circ D$ has a limit in $\I$. Then $[\A,\I]$ has all limits of shape $\mathbf{I}$ and they are preserved (and jointly reflected) by the evaluation functors.
His only justification for this is that $[\A,\I]$ is guaranteed to be locally small under these conditions. Why this is necessary is not expounded upon. My own best guess for this is that the proofs of the above rely on evaluation functors, and if $[\A,\I](X,Y)$ is a large class for some $X,Y$, then $\ev_A$ runs the risk of embedding a large class of natural transformations into a set of arrows $\I(X(A),Y(A))$ which cannot work. That said, nlab claim this theorem $(\ast)$ holds for presheaf categories regardless of whether or not $\A$ is small (it need only be locally small, apparently).
So, which is it? Can I really assume that $(\ast)$ holds if $\A$ is relaxed to be locally small?
I have two motivations for this question. The first is general interest (since the pointwise limit theorem is very powerful), and the second is in the following exercise:
Let $\A$ be a locally small category and $B\in\A$. Show that representables have the following connectedness property: if there exist $X,Y\in\psh_\A$ such that $\A(-,B)\cong X+Y$, exactly one of $X,Y$ are the degenerate constant empty functors.
My solution needs the pointwise limit theorem to apply even for $\A$ locally small:
Modifying $X,Y$ up to isomorphism we have that $\A(A,B)=(X+Y)(A)=X(A)+Y(A)$ for all $A\in\A$ by the pointwise limit theorem, and $(Xp)(f)=f\circ p$ for all $p\in\A(A',A)$, $f\in X(A)\subseteq\A(A,B)$; the same holds true for $Y$.
As $\A(B,B)$ contains $1_B$, always, without loss of generality suppose $1_B\in X(B)$. For any $A\in\A$, if there exists $p\in Y(A)$ then $-\circ p:\A(B,B)\to\A(A,B)$ is equal to the action $Xp:X(B)\to X(A)$, so in particular $1_B\circ p=p\in X(A)$ despite $p\in Y(A)\implies p\notin X(A)$, a contradiction. Therefore there never exists a $p\in Y(A)$, for any $A\in\A$, so $Y$ is the constant empty functor.
Best Answer
There are several "size" issues involved here. One is literally about size: some of the collections involved have too many elements to be small. But another is more about complexity: some collections have a small number of elements but those elements themselves are not small – such collections do not even exist in the ontology of NBG or MK, in the same way that classes do not really exist in the ontology of ZFC. Complexity issues can be resolved by using a set theory with a richer ontology, such as ZFC + universes, but size issues in the narrow sense cannot be worked around so easily.
To be concrete, I will work in ZFC + universes, but what I say applies more generally. Let $\mathcal{A}$ be a category, let $\textbf{Set}$ be the category of small sets, and let $\textbf{Psh} (\mathcal{A})$ be the category of all functors $\mathcal{A}^\textrm{op} \to \textbf{Set}$. If $\mathcal{A}$ is small, then functors $\mathcal{A}^\textrm{op} \to \textbf{Set}$ can be implemented as small sets, and likewise natural transformations between them, so $\textbf{Psh} (\mathcal{A})$ is a locally small category of similar complexity as $\textbf{Set}$. If $\mathcal{A}$ is essentially small, then $\textbf{Psh} (\mathcal{A})$ is equivalent to a locally small category, but possibly not a locally small category in the strictest sense, for complexity reasons. If $\mathcal{A}$ is not even essentially small, then $\textbf{Psh} (\mathcal{A})$ may fail to be locally small even up to equivalence.
Anyway, for any category $\mathcal{C}$ whatsoever:
Proposition. Let $X : \mathcal{I} \to [\mathcal{A}^\textrm{op}, \mathcal{C}]$ be a diagram. If, for each object $A$ in $\mathcal{A}$, the diagram $\textrm{ev}_A \circ X : \mathcal{I} \to \mathcal{C}$ has a limit (resp. colimit), then $X : \mathcal{I} \to [\mathcal{A}^\textrm{op}, \mathcal{C}]$ has a limit (resp. colimit) and, for all objects $A$ in $\mathcal{A}$, $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves that limit (resp. colimit).
The proof is straightforward and constructive (provided the hypotheses are understood constructively, i.e. we are given a (co)limit (co)cone for each $\textrm{ev}_A \circ X$ and not mere existence). Thus it is robust and there are basically no size issues involved. Where size issues (in the narrow sense) start to get involved is when we are less careful about balancing the hypotheses and the conclusions. For example:
Proposition. If $\mathcal{C}$ has copowers (resp. powers) indexed by small sets and $\mathcal{A}$ is locally small, then for all objects $A$ in $\mathcal{A}$, $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves limits (resp. colimits) for all (not necessarily small) diagrams.
Notice that the proposition concludes something about arbitrary limits in $[\mathcal{A}^\textrm{op}, \mathcal{C}]$ only assuming hypotheses on small copowers in $\mathcal{C}$. How can this be? The answer is that the proof uses a trick: it constructs a left adjoint for $\textrm{ev}_A$, and we know that right adjoints have extremely strong limit preservation properties.
Moreover, if you examine the proof you will realise that the size hypotheses are not optimal: it suffices that $\mathcal{C}$ have copowers indexed by the hom-sets of $\mathcal{A}$. So, for example, if $\mathcal{A}$ is a locally finite category and $\mathcal{C}$ has finite copowers then $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves arbitrary limits. Or if $\mathcal{A}$ is a preorder category and $\mathcal{C}$ has an initial object then $\textrm{ev}_A : [\mathcal{A}^\textrm{op}, \mathcal{C}] \to \mathcal{C}$ preserves arbitrary limits.