colimit of functors for a cocomplete codomain
1. Proposition
Let 
be categories, 
the functor category,
a diagram, such that for each
there exists a colimit 
.
Then the colimit in is
where 
is the restricted colim functor for the diagrams
2. Proof
2.1. welldefined
follows from the functorality of the colimit functor. Note that by assumption, there exist enough colimits.
2.2. universal property
Suppose that there exists a functor 
and natural transformations
Then for each component 
, it follows, that
commutes.
Hence by universal property of 
there exists a unique morphism
Let
2.2.1. welldefined
Let
Then by assumption, the diagram above commutes, hence also
commutes.
Furthermore, there exists a unique morphism
making the diagram commute
Hence by uniqueness of
we conclude, that bottom half
commutes.
Therefore, we have a welldefined natural transformation between 
and 
2.2.2. uniquness
Suppose we have two natural transformations 
.
Then for each component
is a commutative diagram for 
resp. 
Hence we conclude by universal property of 
, that 
is unique, thus
Since 
was arbitrary, it follows, that
