Theorem: Fix a small category $\mathbf{A}$. Let $\mathscr{S}$ be a locally small category with pullbacks. For functors $X, Y: \mathbf{A} \rightarrow \mathscr{S}$, a natural transformation $\alpha : X \rightarrow Y$ is monic in $[\mathbf{A}, \mathscr{S}]$ if and only if $\alpha_A$ is monic for all $A$.

Proof: Suppose $\alpha_A$ is monic for all $A$. Then, suppose we have $\eta, \varepsilon : X' \rightarrow X$ so that $\alpha \circ \eta = \alpha \circ \varepsilon$. Then $\alpha_A \circ \eta_A = \alpha_A \circ \varepsilon_A$ for all $A$. But then $\eta_A = \varepsilon_A$ for all $A$ and $\eta = \varepsilon$. So $\alpha$ is monic.

Now suppose that $\alpha$ is monic as a map in $[\mathbf{A}, \mathscr{S}]$. Then, the following square is a pullback:

The evaluation functor preserves limits, so we have that:

is a pullback for each $A$. Suppose we have maps $f, g: J \rightarrow X(A)$ with $\alpha_A \circ f = \alpha_A \circ g$. Then $(J, f, g)$ is a cone over $X(A) \xrightarrow{\alpha_A} Y(A) \xleftarrow{\alpha_A} X(A)$. But then the pullback square above implies that $f = g$, and $\alpha_A$ is monic.