TITLE: Comment about type theory in Lawvere and Rosebrugh's Sets for Mathematics
QUESTION [8 upvotes]: The Foreword to Sets for Mathematics (second section, titled Organization) contains the following comment, about the differences between ETCS and other foundations of mathematics:

Each map needs both an explicit domain and an explicit codomain (not
  just a domain, as in previous formulations of set theory, and not just
  a codomain, as in type theory).

I understand the remark about set theory: a function as a set of ordered pairs determines its domain but not its codomain.
But what do the authors mean when they say a function in type theory has an explicit codomain but not an explicit domain? 
A term of the simple type $A \rightarrow B$ has both an explicit domain and codomain. A term of a dependent type $\Pi x:A.B(x)$ arguably has an explicit domain but not a codomain (since its codomain would need to be something like $\bigcup x:A.B(x)$, which does not exist as a type in Martin-Löf Type Theory).

REPLY [4 votes]: I believe the point is that a term $x:A$ of type theory is interpreted categorically as an arbitrary morphism into $A$.