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Im supposed, in the category $\textbf{Mon}$ of monoids, prove that all equalisers and finite products exist. Does this mean that I have to prove that every pair of arrows has an equaliser, that the category has a terminal object and that the category has binary products?

In my book, I read that a terminal object is a product of no objects, can someone explain this? IS this why we require a terminal object in proving that a category has ALL finite products?

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  1. Yes, it does mean that.
  2. Imagine finite sets of numbers for analogy.
    Define the product of the set $\{a_1,a_2\dots,a_n\}$ as $a_1a_2\dots a_n$.
    What should we mean by the empty product, i.e. the product of $\emptyset$?
    If it will have a value, we have to define it as $\bf1$ to keep the properties, as $1$ doesn't affect the product: $1\cdot x=x$ for any number $x$.

The same goes on for the categorial product, of which the terminal object $\bf1$ is the unit, in that ${\bf1}\times X\cong X$ for any object $X$.

Having all finite products means that the product exists for all finite sets of objects.
$\emptyset$ is a finite set of objects.

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  • $\begingroup$ so you mean that in a category, the product of no objects is defined to be any terminal object (or A terminal object since they are all isomprhic)? Is this a definition, and if so, what motivates it? Thanks for your answer. $\endgroup$ – user117449 Oct 31 '16 at 9:52
  • $\begingroup$ Yes, exactly. Note that any limit is defined only up to equivalence, by a universal property. What motivates it? Just put the empty set of given objects into the definition. $\endgroup$ – Berci Nov 27 '16 at 22:17

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