In Page 387 of the book "Practical Foundations of Mathematics" (by Taylor) it is said that
"colimits with respect to total functions work for partial functions"
as a consequence of the existence of a right adjoint for the forgetful functor from the category of "sets with total mappings" to the category of "sets with partial mappings". See here for the concrete formulation (it is item c).
I find this sentence not very precise so I would like to ask something about this statement just to see whether I have understood it well.
Does it mean that for "every (finite) diagram of sets with total mappings", the colimit in the category of "sets with partial mappings" exists and it is exactly the colimit in the category of "sets with total mappings"?
The highlighted statement means that you can use the exact same construction for constructing the limit, whether you take total functions as morphisms, or only partial functions.