How do the "limits are constructed objectwise thus a property about limits true in $\rm Set$ is also true in $ {\rm{Set}}^I$" argument works? For example, I encountered the following two arguments:
Since sifted colimits and finite products commute in $\rm Set$, they do so in ${\rm Set}^T$ (where they are computed objectwise) where $T$ is an algebraic theory.
Sifted colimits commute with finite products in Alg $T$ since Alg $T$ is closed under limits and sifted colimits in ${\rm Set}^T$ , and such limits and colimits in ${\rm Set}^T$ are formed objectwise.
I am able to understand these statements if I completely flesh out the argument (by considering a diagram from product of finite discrete and sifted category in Set) and I understand this refers to $(\lim D)(i) \cong \lim D(i)$, but I am not quite sure how to use the object-wise statement, to argue like above.
It seems similar to the "$f$ is continuous iff its components are continuous.", however, in that case, we only invoke this to pass around continuity from components to $f$, and not to pass any property, that would be true of the components. If we wanted to pass around more properties, like differentiability, we would first prove a similar variant (perhaps with some more premises) of the above claim and then pass the differentiability from the components to the function itself.