Initial functor

A functor $F:\mathcal{I}\to \mathcal{J}$ is initial when every comma category $(F\downarrow j)$ is nonempty and connected. Restriction along an initial functor preserves limits:

A functor $F:I\to J$ is initial if restriction preserves categorical limits for diagrams in a category with codomain ${\mathbf{Set}}^{\mathrm{op}}$. To test necessity, view $J(-,j)$ as such a diagram. Its limit is a singleton, since the colimit of the corresponding representable presheaf is the connected-component set of the slice $J\downarrow j$, which has a terminal object. The restricted colimit is the connected-component set of $(F\downarrow j)$. Thus that comma category is nonempty and connected. Sufficiency follows from cone restriction along an initial functor.

For an initial functor $F:\mathcal{I}\to \mathcal{J}$, restriction gives an isomorphism between the category of cones over $D:\mathcal{J}\to \mathcal{C}$ and that over $DF$. Given a cone over $DF$, choose $(i,u:Fi\to j)$ and define its $j$-leg as $D(u){\gamma }_i$; connectedness of $(F\downarrow j)$ makes the result independent of the choice.