The real question here is what properties the language of category theory captures.
A statement in the (finitary) language of category theory is one formed from propositions of the form $f = g$ where $f, g : X \to Y$ using the propositional operations of $\lor, \land, \top, \bot, \neg, \implies$, together with quantifiers either over all objects (eg $\forall X$, $\exists X$ where $X$ is an object variable) or over all morphisms between two given objects (eg $\forall f : X \to Y$ or $\exists f : X \to Y$ where $X, Y$ are objects).
Two things can be shown about statements in the language of category theory. Suppose given a statement $\phi(X_1, ..., X_n, f_1, ..., f_m)$ in the language of category theory with object variables $X_1, ..., X_n$ and free function variables $f_i : X_{d_i} \to X_{c_i}$ for $1 \leq i \leq m$.
Now consider a category and two different variable assignments - one assignment $X_1 \mapsto W_1, ..., X_n \mapsto W_n$ where each $W_i$ is an object, and $f_1 \mapsto g_1, ..., f_m \mapsto g_m$ where $g_i : W_{d_i} \to W_{c_i}$ for all $i$, and another assignment $X_1 \mapsto Y_1, ..., X_n \mapsto Y_n$ and $f_1 \mapsto h_1, ..., f_m \mapsto h_m$, where $Y_i$ is an object for all $i$ and $h_j : W_{d_j} \to W_{c_j}$ for all $j$. Suppose there are isomorphisms $k_1 : W_1 \to Y_1, ..., k_n : W_n \to Y_n$ such that for all $i$, $h_i = k_{c_i} \circ f_i \circ k_{d_i}^{-1}$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m) \iff \phi(Y_1, ..., Y_n, h_1, ..., h_m)$. This is known as "isomorphism invariance of truth".
Now consider a functor $F : C \to D$ which is fully faithful and essentially surjective, and a variable assignment in $C$ of the form $X_i \mapsto W_i$, $f_j \mapsto g_j$ as above. We assume that $\phi$ has no free variables other than the $X_i$ and $f_i$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m)$ iff $\phi(F(W_1), ..., F(W_n), F(g_1), ..., F(g_m))$. This is known as "equivalence invariance of truth".
Both of the above can be proved by induction on formulas.
The two above statements can be generalised to the infinitary language of category theory, which allows quantification over external sets, though some care must be taken in settings without the axiom of choice.
So the language of category can only discuss properties which are invariant up to isomorphism and up to equivalence of categories. No finer-grained discrimination is possible.
For example, consider the category of sets. It follows from isomorphism invariance that the language of category cannot distinguish two sets which have the same cardinality. The language of category cannot distinguish a monic $f : S \to T$ from a subset $S \subseteq T$.
For the category of topological spaces with morphisms the continuous maps quotiented by homotopy, the language of category theory cannot distinguish between homotopy equivalent spaces.
So the key when using category theory is to find the right amount of data for morphisms to carry based on the underlying subject one is studying and the particular context.
Best Answer
One of perhaps many perspectives is the following:
This is essentially a 2-categorical analog of a slice/comma category.
In this case, it is the full 1-subcategory of the colax slice 2-category $\newcommand\Set{\mathbf{Set}}\newcommand\Cat{\mathbf{Cat}}\newcommand\C{\mathcal{C}}\Cat/\C$ on the objects which are sets regarded as small discrete categories.
I'll rewrite the definition of the colax slice 2-category here (particularly as it's not explicitly spelled out on the nlab page):
Let $\newcommand\B{\mathcal{B}}\B$ be a 2-category. Let $X$ be an object of $\B$. The colax slice 2-category $\B/X$ is defined to have
0-cells: $\B$-morphisms $a:A\to X$.
1-cells: If $a:A\to X$ and $b:B\to X$ are 0-cells, then a 1-cell $(f,\phi):a\to b$ is a pair of a $\B$-morphism $f:A\to B$ and a $\B$-2-cell $\phi : a\to bf$.
The composition of $(f,\phi):a\to b$ and $(g,\psi):b\to c$ is the pair $$(gf : A\to C, (\psi\operatorname{.} f)\phi : a\to bf \to cgf).$$
2-cells: If $(f,\phi),(g,\psi): a\to b$, a 2-cell $\alpha : (f,\phi)\to (g,\psi)$ is a 2-cell of $\B$, $\alpha : f\to g$, such that $\psi = (b\operatorname{.}\alpha)\phi$ as $\B$-2-cells from $a \to bg$
In our case
Specializing to our case, $\B=\Cat$, $X=\C$, and we restrict the domains of the $0$-cells to be sets, and then if we forget about 2-cells, we get exactly the construction from the question. Namely, objects are functors $i:I\to \C$, and morphisms $i\to j$ are pairs of a function $f:I\to J$ and a natural transformation $\phi: i\to jf$.
The composition also agrees with the composition in the question.
End note on philosophy and intuition of this perspective
Reading the question, what jumps out at me is that it looks a lot like a slice category or arrow category, since the objects are arrows. The next thought is that it's definitely a slice category, since the codomains are all the same. However the morphisms are different than in the 1-categorical slice category, since the triangles are only required to commute up to a homotopy. My next thought is that this looks exactly like what I'd expect a 2-categorical slice category to be, but we'd best be careful about what directions the homotopies go, since we aren't forcing them to be isomorphisms so I Google lax slice category, and up pops the nlab page linked above.