intrinsic semiconductor
I only think of diffusion current occurring for the following reasons:


A PN junction and depletion region. (Doesn't usually apply to intrinsics, almost by definition, though I suppose one could place an intrinsic next to a doped material as with the PIN diode.)
Graded doping. (Doesn't apply to intrinsics, which are assumed to be uniform and undoped and therefore not having graded doping.)
Carrier generation by some other mechanism in a specific, local area within the material; as in photogenerated carriers. (May apply to intrinsics.)


What's the circumstance you are discussing?

If it is an intrinsic material and there is diffusion currents, it would seem to me that there is some unstated mechanism causing the generation of holes and electrons. Most commonly, I think, that would be due to photogeneration at some spot within  an intrinsic.

It might be in the case of a PIN diode.

Maybe someone better informed than I am can help. But that's as far as I can get with my limited knowledge and your vague description.