Assuming Choice, precomposition with a final functor does not change colimits, and precomposition with an initial functor does not change limits

Restriction sends a cocone ${\rho }_b:F(b)\to X$ to ${\rho }_{u(a)}:F(u(a))\to X$. Conversely, from a cocone $\lambda$ over $Fu$, use [F1] and [F3] to choose for each $b$ an object $(a_b,{\beta }_b:b\to u(a_b))$ and put ${\bar{\lambda }}_b={\lambda }_{a_b}F({\beta }_b)$.

If $h:(a,\beta )\to (a^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }})$ is a comma morphism, then $u(h)\beta ={\beta }^{\mathrm{\prime }}$ and the cocone equation gives ${\lambda }_{a^{\mathrm{\prime }}}F({\beta }^{\mathrm{\prime }})={\lambda }_{a}F(\beta )$. A finite zigzag therefore proves that ${\bar{\lambda }}_b$ is independent of the chosen comma object.

For $r:b\to b^{\mathrm{\prime }}$, choose a comma object $(a,\beta :b^{\mathrm{\prime }}\to u(a))$ for $b^{\mathrm{\prime }}$. Then $(a,\beta r)$ is one for $b$, so step 1.2 gives ${\bar{\lambda }}_b={\bar{\lambda }}_{b^{\mathrm{\prime }}}F(r)$. Thus $\bar{\lambda }$ is a cocone over $F$.

Restricting $\bar{\lambda }$ at $u(a)$ and using $(a,1_{u(a)})$ returns ${\lambda }_a$. Extending the restriction of $\rho$ returns ${\rho }_b$ by its cocone equation. Hence restriction is a bijection between cocones over $F$ and over $Fu$, natural in their apex.

By [F2], an initial object represents either naturally identical cocone assignment exactly when it represents the other. Thus either colimit exists if and only if the other does. When both are chosen, [L1] identifies the induced canonical map as their unique compatible isomorphism.

Applying [L2] to steps 1.1 to 2.1 replaces $(b\downarrow u)$ by the comma categories for an initial functor and proves the limit assertion, including both directions of existence.