Assuming Choice, nonempty sets have all small products but a parallel pair with no equalizer and hence a diagram with no limit

The ordinary Cartesian product of any set-indexed family of nonempty sets is nonempty by [F3] and has the product property [F1] inside the full subcategory. For the empty family, the singleton is a nonempty terminal object. Thus ${\mathbf{S}\mathbf{e}\mathbf{t}}_{\mathrm{\ne }\mathrm{\varnothing }}$ has all small products.

Let $f,g:\{\ast \}\rightrightarrows \{0,1\}$ be the constant maps with values $0$ and $1$. If $h:X\to \{\ast \}$ equalized them, then $fh$ and $gh$ would be the distinct constant functions on the nonempty set $X$. Hence no equalizing cone exists in ${\mathbf{S}\mathbf{e}\mathbf{t}}_{\mathrm{\ne }\mathrm{\varnothing }}$, so in particular no equalizer [F2] exists.

The parallel-pair diagram is finite and small but has no limit, refuting the statement. This is exactly the missing equalizer data isolated by [L1].