  A 1978 theorem of Kozen states that two graphs on $n$ vertices are isomorphic
if and only if there is a clique of size $n$ in the weak modular product
between the two graphs. Restricting to bipartite graphs and considering
complete bipartite subgraphs (bicliques) therein, we study the combinatorics of
the weak modular product. We identify cases where isomorphism is tractable
using this approach, which we call Isomorphism via Biclique Enumeration (IvBE).
We find that IvBE is polynomial for bipartite $2K_2$-free graphs and
quasi-polynomial for families of bipartite graphs, where the largest induced
matching and largest induced crown graph grows slowly in $n$, that is,
$O(\mathrm{polylog }\, n)$. Furthermore, as expected a straightforward
corollary of Kozen's theorem and Lov\'{a}sz's sandwich theorem is if the weak
modular product between two graphs is perfect, then checking if the graphs are
isomorphic is polynomial in $n$. However, we show that for balanced, bipartite
graphs this is only true in a few trivial cases. In doing so we define a new
graph product on bipartite graphs, the very weak modular product. The results
pertaining to bicliques in bipartite graphs proved here may be of independent
interest.
