ยง Why terminal object is a limit
- Thinking in Set, the terminal object is
{*}, which is the empty product of sets. Hence, the terminal is a type of product, which is a limit. - What does the terminal
{*} project onto? It should project onto its components, since its a limit. But recall that it was the limit of ZERO objects. {*} vacuously projects into zero objects [ie, we're not obliged to construct a projection ] - Think about products again. the product is universal such that any other "candidate for the product" must factor through a projection onto the product. Similarly, the terminal is universal such that any other "candidate for the terminal" (literally all other objects) must factor through the terminal [ie, must map into the terminal ].