The accepted answer is good. If you would like another reference see section 2.15 of the Handbook of categorical algebra, volume I (edited by F. Borceux) pages 87--90. In particular, their Corollary 2.15.3 tells us the following:
Let $F,G: \mathcal{C} \rightarrow\mathcal{D}$ be two functors where $\mathcal{C}$ is a small category and $\mathcal{D}$ has pullbacks. Then a natural transformation $\alpha : F \rightarrow G$ is monic in $\mathcal{D}^{\mathcal{C}}$ if and only if for each object $C\in \mathcal{C}$, $\alpha_C : F(C)\rightarrow G(C)$ is monic in $\mathcal{D}$.
