at.algebraic topology - Principle when limits level by level don't commute with simplicial structure

Limits and colimits can be computed point-wise in functor categories, such as the category of simplicial objects of any category. That is, if you have a functor category $mathcal{C}^I$ ($I = Delta$ in your case) and a functor $F: P longrightarrow mathcal{C}^I$, in particular, for every object $p$ in $P$ you have a functor $F_p : I longrightarrow mathcal{C}$. Then $varinjlim_p F_p$ is an object of $mathcal{C}^I$; that is, a functor $I longrightarrow mathcal{C}$, whose value on objects $i$ in $I$ is

$$(varinjlim_p F_p)(i) = varinjlim_p (F_p(i)) .$$

This is true as far as the colimit on the right exists for every object $i$ in $I$. An analogous statement applies for limits in functor categories.

For limits, you can find the result in MacLane, "Categories for the working mathematician", chapter V, section 3, theorem 1 (see at the end of the proof also). For colimits, see "Sheaves in geometry and logic: a first introduction to topos theory", by Saunders Mac Lane,Ieke Moerdijk, p.40.

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