  We study isomonodromicity of systems of parameterized linear differential
equations and related conjugacy properties of linear differential algebraic
groups by means of differential categories. We prove that isomonodromicity is
equivalent to isomonodromicity with respect to each parameter separately under
a filtered-linearly closed assumption on the field of functions of parameters.
Our result implies that one does not need to solve any non-linear differential
equations to test isomonodromicity anymore. This result cannot be further
strengthened by weakening the requirement on the parameters as we show by
giving a counterexample. Also, we show that isomonodromicity is equivalent to
conjugacy to constants of the associated parameterized differential Galois
group, extending a result of P. Cassidy and M. Singer, which we also prove
categorically. We illustrate our main results by a series of examples, using,
in particular, a relation between Gauss-Manin connection and parameterized
differential Galois groups.
