I am trying to show that right adjoints preserve limits. So suppose we have $F: C \rightarrow D$ and $G: D \rightarrow C$ be 2 adjoint functors with $G$ the right adjoint. So suppose we have a limit $(N, \phi)$ of the diagram $F: C \rightarrow D$ where $\phi$ is indexed by objects $X$ in $C$ with $\phi_X: N \rightarrow F(X).$ Now I want to show that $(G(N), G(\phi))$ is a limit of $G: D \rightarrow C.$ However, something that confuses me is that $G(\phi_X): G(N) \rightarrow G(F(X))$ are the morphisms but can this be indexed by every object in $D?$ The family of morphisms for a cone for the object $G(N)$ should be $\psi: G(N) \rightarrow G(Y)$ for every object $Y$ in $D.$ So does $F(X)$ hit every object in $D?$ That is, for every object $Y$ in $D$ does there exist $X$ in $C$ such that $F(X) = Y?$
Do we need every object to be hit?