For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise

For $h:a\to b$, the maps $D(j)(h):D(j)(a)\to D(j)(b)$ form a natural transformation of $\mathcal{J}$-diagrams. By [L1] they induce $L(h):L(a)\to L(b)$ with ${\lambda }_j^{b}L(h)=D(j)(h){\lambda }_j^a$.

Given a cone $\xi :X\Rightarrow D$ in the functor category, pointwise universality gives a unique $u_a:X(a)\to L(a)$ with ${\lambda }_j^a{u}_a={\xi }_{j,a}$. For $h:a\to b$, naturality of ${\xi }_j$ makes $L(h)u_a$ and $u_{b}X(h)$ equal after every ${\lambda }_j^b$; [L2] makes them equal. Thus the $u_a$ form a natural transformation $u:X\Rightarrow L$.

Identity and composition for $L$ follow either from [L1] or by composing with every ${\lambda }_j^a$ and applying [L2]. Thus $L:\mathcal{A}\to \mathcal{C}$ is a functor, and the displayed equations say each ${\lambda }_j:L\Rightarrow D(j)$ is natural.

The transformation $u$ factors the cone componentwise. Any other factor has the same component at every $a$ by pointwise uniqueness, hence equals $u$. By [F2], $(L,\lambda )$ is a limit in the functor category.

If only existence, rather than chosen limits, is assumed, [F3] selects the pointwise cones over the set of objects of the small category $\mathcal{A}$. Reversing the whole construction by [L3] proves the colimit assertion.