Let $\mathbb I$ be a category with exactly $2$ objects and $4$ arrows. The $2$ arrows that are not identities are parallel and their (common) domain and (common) codomain are distinct. Looking at the category $$\mathsf Set^{\mathbb I}$$ (having as objects the functors $\mathbb I\rightarrow\mathsf Set$ and as arrows the natural transformations) I wondered if this category can be recognized as the category of (small) graphs. Is that indeed the case, or am I overlooking something?
If I am right here then the fact that $\mathsf Set$ is (co)complete justifies the the conclusion that $\mathsf Grph$ is (co)complete.