What kind of entity is the thing that sends a category to the initial object of said category?
Take for example a group presentation like $\langle a, b, c \mid ab = ba \rangle$. The group that this presentation refers to is the initial object in the category of models of an equational theory with signature $(1, a, b, c; \bigcirc^{-1}; *)$ and equations consisting of the group axioms and $ab = ba$.
Similarly, $\mathbb{Z}$ is the initial object in the category of rings.
It makes sense to think of the "get the initial object" operation some type of (partial) function-like entity:
We can even give it a type using some dependent-type-like notation:
$$ I : \left(\Pi\; C : \mathrm{Cat} \mathop. C_\text{obj}\right) $$
$I$ has at least one nice property as a function-like entity:
If $C$ and $D$ are both categories and $C \times D$ is the product category.
Then $I(C \times D)$ is the object $(I(C), I(D))$, when defined.