I am self-reading An Introduction To The Language Of Category by Roman and having some questions about the set-up for proving the existence of limits.
In the book, the hom-classes are assumed to be sets. A screenshot of the book (parts of page 109 & 110) regarding my questions is attached at the end of the post.
My first question is "the product of a small diagram" (The 2nd sentence in the first paragraph). The product is defined for a diagram with no morphisms (except for identities). Then how to define the product of a small diagram? Do we ignore all morphisms other than identities?
The second question is the finiteness of in-degree $m_k$ (In the final paragraph). We only assume $\mathbb{D}$ to be a small diagram, so for $X\in $ Ob$(\mathbb{D})$, Hom(X,X) can be a countably infinite set. Then the in-degree may be infinite.
Finally, if $X\in $ Ob$(\mathbb{D})$ has no arrow entering $X$, then the in-degree should be counted as 1, right? (i.e. we consider identity when counting in-degree) Then $D_2$ in Figure 61 should have 3 copies instead of 2?
