Adding Links to Category Images

try something like this:

<?php
foreach((get_the_category()) as $category) {
  $html = '<a href="' . get_category_link($category->cat_ID) . '">';
  $html .= '<img src="https://example.com/images/' . $category->cat_name . '.jpg" alt="' . $category->cat_name . '" />';
  $html .='</a>';
  echo $html;
}
?>

This works like a charm. I know I’m stretching it, but is there a way for the Category name to show under each image?

You could do something like this (not tested):

<?php
foreach((get_the_category()) as $category) {
  $html = '<p class="cat-img">';
  $html .= '<a href="' . get_category_link($category->cat_ID) . '">';
  $html .= '<img src="https://example.com/images/' . $category->cat_name . '.jpg" alt="' . $category->cat_name . '" />';
  $html .='</a><span>' . $category->cat_name . '</span></p>';
  echo $html;
}
?>

and put this at the bottom of your theme’s style.css:

.cat-img a {
	float: left;
}
.cat-img span {
	clear:  left;
}

Thanks! Here’s what I ended up doing for it to work.

To display the images (categories) with the category names underneath:

<div align="center"><?php
foreach((get_the_category()) as $category) {
  $html = '<div class="cat-img">';
  $html .= '<a href="' . get_category_link($category->cat_ID) . '">';
  $html .= '<img src="https://www.yoursite.com/images/' . $category->cat_name . '.jpg" alt="' . $category->cat_name . '" />';
  $html .='</a><br>' . $category->cat_name . '</div>';
  echo $html;
}
?></div>

For the css:

.cat-img {
	float:left;
width:300px;
text-transform:uppercase;
padding:15px;
}

The width above is equal to the size of the images.