In higher category theory
The statement of the Yoneda lemma generalizes from categories to (∞,1)-categories.
(HTT, prop. 188.8.131.52)
For a small -category and an -functor, the composite
is equivalent to .
(HTT, Lemma 184.108.40.206)
Preservation of limits
The -Yoneda embedding preserves all (∞,1)-limits that exist in .
(HTT, prop. 220.127.116.11)
Local Yoneda embedding
For an (∞,1)-site and an (∞,1)-topos, (∞,1)-geometric morphisms from the (∞,1)-sheaf (∞,1)-topos to correspond to the local (∞,1)-functors , those that
More preseicely, the (∞,1)-functor
given by precomposition of inverse image functors by ∞-stackification and by the (∞,1)-Yoneda embedding is a full and faithful (∞,1)-functor and its essential image is spanned by these local morphisms.
(HTT, prop. 18.104.22.168)
See also the discussion on MathOverflow.