What does the sentence "[...] these statements should be interpreted, of course, in the internal language of the topos $\mathcal{E}$" mean, in the context of, say, the definition of a groupoid in $\mathcal{E}$ as a graph satisfying equations and endowed with an associative composition?
I am only familiar with Grothendieck topoi. Does that statement have something to do with the logic aspects of elementary topoi?