title: How is "a monoid on applicative functors" different than "a monoid in the category of endofunctors"?
body: <p>Perhaps neither of these statements are categorically precise, but a monad is often defined as "a monoid in the category of endofunctors"; a Haskell <a href="https://hackage.haskell.org/package/base-4.11.1.0/docs/Control-Applicative.html#t:Alternative" rel="noreferrer"><code>Alternative</code></a> is defined as "a monoid on applicative functors", where an <a href="https://hackage.haskell.org/package/base-4.11.1.0/docs/Control-Applicative.html" rel="noreferrer">applicative functor</a> is a "strong lax monoidal functor". Now these two definitions sound pretty similar to the ignorant (me), but work out significantly differently. The neutral element for alternative has type <code>f a</code> and is thus "empty", and for monad has type <code>a -&gt; m a</code> and thus has the sense "non-empty"; the operation for alternative has type <code>f a -&gt; f a -&gt; f a</code>, and the operation for monad has type <code>(a -&gt; f b) -&gt; (b -&gt; f c) -&gt; (a -&gt; f c)</code>. It seems to me that the real important detail is <em>in the category of endofunctors</em> versus <em>over endofunctors</em>, though perhaps the "strong lax" detail in alternative is important; but that's where I get confused because within Haskell at least, monads end up being alternatives: and I see that I do not yet have a precise categorical understanding of all the details here.</p>
  
  <p>How can it be precisely expresseed what the difference is between alternative and monad, such that they are both monoids relating to endofunctors, and yet the one has an "empty" neutral and the other has a "non-empty" neutral element?</p>