More generally, let and be arbitrary categories. The -ary diagonal functor of is the functor sending each object to the constant functor (the functor having value for each object of and value for each arrow of ), and each arrow of to the the natural transformation which has the same value at each object of .
Since is -cocomplete (-complete) iff has a left (right) adjoint, the general adjoint functor theorem may be used in some cases to prove cocompleteness (completeness). For this to work, must at least preserve small limits (colimits).
Let and be arbitrary categories. Then preserves all limits that exist in .
First, recall that limits in functor categories are calculated pointwise. In some detail, if for an object we write for the ”evaluate at ” functor (with and ), then we have the following fact (Theorem V.3.1 on p. 115 of Categories Work): If is such that for each object of , has a limiting cone , then there exists a unique functor with object function such that with is a cone ; moreover, this is a limiting cone from to .
Back to the proof of the proposition, let be a functor with a limiting cone . We would like to show that is a limiting cone. Noting that (where the first is and the second is ), the last cone may be written as .
First, we note that for each object of , is just , and therefore has the limiting cone by assumption. Hence, it is clear that has a limit, but we must verify that is a limiting cone.
One functor with object function is just . For this functor, we have our cone . Since for all and we have , we are done by the theorem on pointwise limits.