I was reading about limits and colimits from May's A Concise Course in Algebraic Topology (Chapter 2, Sec 6). He defines the limits essentially like so:
Let $\mathsf I$, $\mathsf C$ be categories and $F\colon \mathsf{I\to C}$ be a functor (a.k.a. an $\mathsf I$-shaped diagram in $\mathsf C$). Then a limit of $F$ is an object $\lim F$ together with a natural transformation $\pi\colon \underline{\lim F}\Rightarrow F$ (see the note below) such that any other functor of the form $\varepsilon\colon\underline{A}\Rightarrow F$ factors through $\pi$ via a unique morphism $A\to\lim F$ (more precisely, through the corresponding natural transformation $\underline A\Rightarrow \underline{\lim F}$ (again see the note)).
Note: In the above, for any object $A$ of $\mathsf C$, $\underline A$ denotes the constant functor $\mathsf{I\to C}$ that maps each object to $A$ and each morphism to $1_A$. Any morphism $f\colon A\to B$ in $\mathsf C$ induces a natural transformation $\underline f\colon \underline A\Rightarrow \underline B$ such that $\underline f_i = f$ for each object $i$ of $\mathsf I$.
This definition screams to me to be postulated so that limits become the terminal objects in some appropriate category. I attempt to define that category and request any criticism/validation.
Attempt at the definition. Given a functor $F\colon\mathsf{I\to C}$, a limit of $F$ is simply a terminal object in the category whose objects are natural transformations of the form $\underline A\Rightarrow F$ and morphisms are commutative diagrams of the following form:
Compositions are defined via composition of natural transformations.
What do you think?
This is correct. To put this in standard terminology:
Let $F : I \to C$. Define the diagonal functor $\Delta : C \to C^I$ by sending each $A$ in $C$ to $\underline{A}$, with the obvious action on morphisms. Then a limit of $F$ is simply a universal arrow from the functor $\Delta$ to $F$ in $C^I$. This is equivalently a terminal object in the comma category $\Delta \downarrow \underline{F}$.
Explicitly, an object of the comma category is a pair of $c$ in $C$ along with an morphism $\lambda: \Delta (c) \to F$, which is a natural transformation. To say that this is a terminal object is to say that, for any other $(c' , \alpha)$, there exist a unique $f : c' \to c$ such that $\lambda \cdot \Delta(f) = \alpha$.
A dual statement can be formulated for colimits.