  In this article, we construct bipartite graphs which are cospectral for both
the adjacency and normalized Laplacian matrices using partitioned tensor
product. This extends the construction of Ji, Gong, and Wang
\cite{ji-gong-wang}. Our proof of the cospectrality of adjacency matrices
simplifies the proof of the bipartite case of Godsil and McKay's construction
\cite{godsil-mckay-1976}, and shows that the corresponding normalized Laplacian
matrices are also cospectral. We partially characterize the isomorphism in
Godsil and McKay's construction, and generalize Ji et al.'s characterization of
the isomorphism to biregular bipartite graphs. The essential idea in
characterizing the isomorphism uses Hammack's cancellation law as opposed to
Hall's marriage theorem used by Ji et al.
