Weighted limits in ordinary category theory are ordinary limits from the category of elements

Question

In this question, the OP says that

In $\mathsf{Set}$-enriched category theory, one can say that the limit of $\mathbf{J} \xrightarrow{D} \mathscr{A}$ weighted by $\mathbf{J} \xrightarrow{W} \mathsf{Set}$ can be equivalently expressed as an ordinary limit of $D$ precomposed with the projection functor from the elements of $W$.

While I believe and understand the statement, I would like a reference for it, with some context. Can anyone point me to one?

Answer 1

You can find detailed explanations in either

• Section 3.4 of Kelly's "Basic Concepts of Enriched Category Theory", available here.
• Section 7.1 of Riehl's "Categorical Homotopy Theory", available here.
• Proposition 4.1.9 of Loregian's "Coend calculus", available here.