$$\newcommand{\cSet}{\mathsf{Set}}$$ $$\newcommand{\Hom}{\operatorname{Hom}}$$
I'm trying to calculate the action of a limit of functors on morphisms. Emily Riehl's "Category Theory in Context" says:
The universal property of the limit construction is used to define the action of this functor on morphisms.
So I'm trying to figure out how to induce the action of the limit functor on morphisms.
Setup
Let $A \equiv ( 0 \xrightarrow{f} 1 )$ be a category, and let's consider the functor category $[A, \cSet]$. I wish to compute the limit of the diagram category $D \equiv (x \xrightarrow{g} y \xleftarrow{h} z)$ in the functor category $[A, \cSet]$. More precisely, suppose I have a diagram given by $K: D \to [A, \cSet]$. Let's call $K(x) = X$, $K(y) = Y$, $K(z) = Z$, $K(g) = \gamma$, $K(h) = \delta$. So this looks like:
$$ \begin{matrix} [X0 \xrightarrow{Xf} X1] \Rightarrow_{\gamma} [Y0 \xrightarrow{Yf}{Y1}] \Leftarrow_{\delta} [Z0 \xrightarrow{Zf} Z1] \end{matrix} $$
Calculating the limit
A limit to this diagram is given by a functor $L: [A, \cSet]$ along with three natural transformations $\lambda_x: L \Rightarrow X$, $\lambda_y: L \Rightarrow Y$, $\lambda_z: L \Rightarrow Z$. Let's first calculate the $L0, L1, Lf$.
- We calculate $L_0$ as the limit of the diagram: $X0 \xrightarrow{\gamma_0} Y0 \xleftarrow{\delta_0} Z0$.
- Similarly, we calculate $L_1$ as the limit of the diagram $X_1 \xrightarrow{\gamma_1} Y_1 \xleftarrow{\delta_1} Z_1$.
- Now, how do we calculate $L(f) \in \Hom(L0, L1)$?
- Since $L0$ is the limit of a diagram involving $X0, Y0, Z0$, we have the projections from $L0$ to $X0, Y0, Z0$.
- Next, we have the maps $X0 \xrightarrow{Xf} X1$, $Y0 \xrightarrow{Yf} Y1$, $Z0 \xrightarrow{Zf} Z1$ which allow us to map from $L0$ into $X1, Y1, Z1$.
- If we can prove that this allows us to have a cone with apex $L0$ and base $X1 \xrightarrow{\gamma_1} Y1 \xleftarrow{\delta1} Z1$, we then get a factorization map of the apex of the cone ($L0$) to the apex of the limit ($L1$) which is the map from $L0$ to $L1$ we are looking for.
- How do we show that $L0$ is an apex of the cone with base $X1 \xrightarrow{\gamma_1} Y1 \xleftarrow{\delta1} Z1$?
I'm stuck at this point; I don't really know if I'm doing the right thing in attempting to calculate what $L_f$ is supposed to be.