Product of Beta distribution and scalar between $0$ and $1$

Let $$B(\alpha, \beta)$$ denote the Beta distribution with parameters $$\alpha$$ and $$\beta$$. What do we know about a random variable $$Y \sim cB(\alpha, \beta)$$ where $$c \in (0, 1)$$?

Obviously, $$Y$$ does not follow a Beta distribution anymore, as the support of the distribution of $$Y$$ has to be $$(0, c)$$ now. Does $$Y$$ have any well-known distribution? Is the density function of $$Y$$ just ''squeezed'' into $$(0, c)$$?

Yes, it is just squeezed. It is a variation of non-standard beta distribution, though with non-symmetric bounds. The get the probability density function you just need to "scale back" the variable to unit range, so if $$Y = cX$$, where $$X \sim \mathsf{Beta}(\alpha, \beta)$$, you need to take $$Y/c$$ and pass it through the beta pdf and scale accordingly.