For some motivation I have been wondering about generalizing the topos of coalgebras theorem to relative monads in my previous question. This brought me to wonder about topos objects.

NLab on specifying fully formal ETCS states the following:

For example, the theory of strict toposes is a finite limit theory (it is finitary-algebraic over the category of directed graphs), meaning the notion of strict topos object can be internalized within any finitely complete category.

I am looking for proofs (potentially references) that the theory of strict toposes is indeed a finite limit theory. Or equivalently for the 2 claims, that it is finitary-algebraic over directed graphs and that being finitary algebraic over directed graphs is enough to be finite limit.