Simple question on diagrams in a category

Question

Forgive the simplicity of my question but after running across the definition of a diagram in $C$ of shape $J$ as simply a functor $D:J\rightarrow{C}$ does this require $J$ to be a subcategory of $C$ since the definition is of a diagram of shape $J$ IN $C$

Answer 1

No, for example if your diagram is just a single arrow $A \to B$ then $J$ will be a category with two objects, say $\{1, 2\}$ and one non-identity homomorphism $1 \to 2$. Then the diagram $A \to B$ is given by the functor $J \to C$ which sends $1 \mapsto A$, $2 \mapsto B$, and $(1 \to 2) \mapsto (A \to B)$. Here $C$ could be anything, sets, groups, top, whatever. We can define functors from this $J$ into many different categories to get diagrams of the same shape, and we never have to change $J$.