  A Demazure crystal is the basis at $q=0$ of a Demazure module. Demazure
crystals play an important role in Schubert calculus because the character of a
Demazure crystal in type A is identical to a key polynomial, which is closely
related to Schubert polynomials. In this paper, we study tensor products of
Demazure crystals. Each connected component of a tensor product of Demazure
crystals need not be isomorphic to some Demazure crystal. We provide a
necessary and sufficient condition for every connected component of a tensor
product to be isomorphic to some Demazure crystal. Also, we obtain the explicit
formula for connected components. As applications, we study the positivity for
structure constants of products of key polynomials, and we obtain an equation
of crystals, which is an analog of the Leibniz rule for Demazure operators.
