I think I just made the greatest revelation possible in the automotive world right now: Toyota has a huge gaping hole in their lineup right now, and it's so incredibly easy to plug! What they really need is a great Corolla-sized hybrid, namely in the form of a Corolla Hybrid.
Toyota perennially ranks at the top of hybrid lists, namely through the Prius, so it makes sense to actually expand on that further and add more product to give consumers greater Toyota-branded choice against the competition. The Corolla also happens to be among the best selling cars in the world, so it makes perfect sense to combine the two.
Now, I know what you're thinking. You're thinking a bunch of points, so let me address them individually:
A hybrid Corolla won't eat into Prius sales - The Prius is actually a good step (or at least half-step) up in size (at least by interior volume) than the Corolla, so they're not going to be competing in the same segment regardless of drive train. Toyota has been selling hybrid Camrys for years without fear of cannibalizing sales either from conventionally-powered versions of itself or from the Prius.
A hybrid Corolla won't eat into Prius c sales, either - Once again, different class segments by interior volume, although this time it's the Corolla that's a good step (or half-step) up. If the Prius c is being marketed towards young single people or newly-formed couples, I could see a hybrid Corolla appealing to newly expanded but still small families, families in need of a second car or aging and shrinking families and couples entering into retirement, all of which are seen as major segments of the traditional Corolla market.
Honda has (had) done it for years - by offering exactly a hybrid version of the Civic despite their own Insight (and in the end proving far more popular than the Insight anyway). By folding hybrids into their other model lines instead of marketing a separate model line or sub-brand ala Prius, Honda has managed to both offer consumers a hybrid choice while strengthening their core model lines' brand power and sales figures - a strategy Toyota (and all other manufacturers) have adopted as well.
A hybrid Corolla has legitimate brand-strengthening qualities - The average Corolla buyer typically does not want to stand out from the crowd, hence buying a Corolla in the first place. The Prius sub-brand has garnered a very certain and tremendous amount of brand image (or "good will" as they say in marketing parlance) that has worked great in helping to market that sub-brand - but at the same time, it doesn't necessarily appeal to everyone or even all prospective hybrid buyers. In fact, many consumers are actively turned off by the Prius image.
The Corolla, on the other hand, managed to work out an effective "good will" brand niche along the opposite approach. Many hybrid buyers likely will find the Camry Hybrid (and even the Prius) too large for their needs, and don't really want to buy into the Prius' brand image or even the youthful, hipster approach of the Prius c (or even find the Prius c a tad too small).
A hybrid Corolla allows Toyota to leverage its greater brand strengths - Sure, said prospective buyer can likely be very easily talked into a conventional power train Corolla - or a Ford Focus, or Chevy Sonic, or Hyundai Elantra, or Kia Forte, or hell even a Chevy Trax which if its Encore/Mokka cousin is any indication will be absolutely miserly compared to your typical CUV. A Corolla Hybrid lets Toyota point out that it's still the market leader in technological innovation in this segment, and gives a cutting-edge drivetrain choice few other manufacturers offer in this segment.