The category $Cat$ of small categories is complete. Could you spell out with details the construction of the limit of a functor $F : J \to Cat$ by products and equalizers? (Mac Lane, Categories for the Working Mathematician, Chapter V, Section 2, Theorem 2)
[Math] Limits by Products and Equalizers
category-theory
Best Answer
The limit of a functor $F :B \to C$ can be constructed as the equalizer of
$$s,t :\prod_A FA \longrightarrow \prod_{f : A \to B}FB$$
where $s$ and $t$ are the unique morphisms defined by $$ \pi_{(f : A \to B)}s =\pi_B\\ \pi_{(f : A \to B)}t = (Ff)\pi_A $$ and the universal cone is given by composing with the projections of $\displaystyle\prod\limits_A FA$.
In case $C = Cat$, note that $FA$ is a category for all $A \in B$, and $s$, $t$, and $Ff$ are functors. The functor $s$ is defined on an object $(X_A)_A \in \displaystyle\prod\limits_A FA$ as
$$s((X_A)_A) = (X_B)_{f : A \to B}$$
that is, the $(f : A \to B)$-component of $s((X_A)_A)$ is $X_B$. The functor $t$ is defined on $(X_A)_A$ as
$$t((X_A)_A) = ((Ff)(X_A))_{f : A \to B}$$
On the morphisms of the product category $\displaystyle\prod\limits_A FA$ (these are families of morphisms $g_A : X_A \to Y_A$ in $FA$ for each $A \in B$), the functors $s$ and $t$ are defined by the same formulas:
$$s((g_A)_A) = (g_B)_{f:A\to B}$$
and
$$t((g_A)_A) = ((Ff)(g_A))_{f:A\to B}$$
Finally, note that the limit appears as the subcategory of $\displaystyle\prod\limits_A FA$ consisting of objects and morphisms equalized by $s$ and $t$. Explicitly, an object of the limit $L$ is $(X_A)_A \in \displaystyle\prod\limits_A FA$ such that for all $f : A \to B$ we have $X_B = (Ff)(X_A)$, and a morphism between objects $(X_A)_A$ and $(Y_A)_A$ in $L$ is a family of morphisms $(g_A : X_A \to Y_A)_A$ in $\displaystyle\prod\limits_A FA$ such that $g_B = (Ff)(g_A)$ for all $f : A \to B$.