title: f, g, h :: Kleisli ((->) e) a b <=> f >>> (g &&& h) = (f >>> g) &&& (f >>> h)?
body: <p>Edit: We will call an arrow <code>p</code> pure if exists such function <code>f</code> that: <code>p = arr f</code>.</p>
  
  <p>I'm trying to get a better grasp of Arrows in Haskell, and I want to figure out when </p>
  
  <p><code>f &gt;&gt;&gt; (g &amp;&amp;&amp; h) = (f &gt;&gt;&gt; g) &amp;&amp;&amp; (f &gt;&gt;&gt; h)</code>, where <code>f</code>, <code>g</code>, <code>h</code> are arrows.</p>
  
  <p>Obviously, it isn't generally true. In this particular example, the side-effects are duplicated on the right hand:</p>
  
  <pre><code>GHCi&gt; c = Kleisli $ \x -&gt; ("AB", x + 1)
  GHCi&gt; fst . runKleisli (c &gt;&gt;&gt; c &amp;&amp;&amp; c) $ 1
  "ABABAB"
  GHCi&gt; fst . runKleisli ((c &gt;&gt;&gt; c) &amp;&amp;&amp; (c &gt;&gt;&gt; c)) $ 1
  "ABABABAB"
  </code></pre>
  
  <p>Obviously, <code>f &gt;&gt;&gt; (g &amp;&amp;&amp; h) = (f &gt;&gt;&gt; g) &amp;&amp;&amp; (f &gt;&gt;&gt; h)</code> if <code>f</code> is pure.</p>
  
  <p>I was experimenting in GHCi with this statement for <code>f, g, h :: Kleisli ((-&gt;) e) a b</code>, and didn't manage to find such values of <code>f</code>, <code>g</code> and <code>h</code> that <code>f &gt;&gt;&gt; (g &amp;&amp;&amp; h) ≠ (f &gt;&gt;&gt; g) &amp;&amp;&amp; (f &gt;&gt;&gt; h)</code>. Is this statement indeed true for <code>f, g, h :: Kleisli ((-&gt;) e) a b</code>, and, if so, could this be a valid proof of that:
  The effect of the <code>Monad ((-&gt;) e)</code> is reading from the environment. Thus, the result of the application of <code>f</code> is the function with the help of which <code>g</code> and <code>h</code> are going to read from the environment. No matter where this function was created – it is the same since it is applied to the same argument every time, thus the result of reading from the environment is the same, thus the overall result is the same as well.</p>