JSFiddle - React, Tailwind, and code Playground

HTML

<div class="drop1">
    <select>
        <option value="categ1">Category1</option>
        <option value="1">1</option>
        <option value="2">2</option>
        <option value="3">3</option>
        <option value="4">4</option>
        <option value="5">5</option>
        <option value="6">6</option>
        <option value="7">7</option>
        <option value="8">8</option>
    </select>
</div>
<div class="drop2">
    <select>
        <option value="categ2">Category2</option>
        <option value="1">1</option>
        <option value="2">2</option>
        <option value="3">3</option>
        <option value="4">4</option>
    </select>
</div>

CSS

.drop1, .drop2 {
    float: left;
    width: 180px;
    height: 34px;
    overflow: hidden;
    background: url(images/dropdwn.png) no-repeat right #ccc;
    border-radius: 4px;
    border:1px solid #000;
    box-shadow: 0 1px 3px #f00;
    font-size:14px;
    margin:0 10px;
    position:relative;
}
.arrow {
    position: absolute;
    display: block;
    width: 20%;
    height: 100%;
    background: #0f0;
    top: 0;
    right: 0;
}
.drop1 select, .drop2 select {
    background: transparent;
    width: 100%;
    padding: 5px;
    font-size: 14px;
    border: 0;
    border-radius: 0;
    height: 34px;
    -webkit-appearance: none;
    color: #f00;
    cursor: pointer;
}
.drop1 option, .drop2 option {
    text-align: center;
    margin: 0 auto;
    color: #000;
    cursor: pointer;
}

JavaScript

$(".drop1, .drop2").append("<div class='arrow' />");

    $(".ie-arrow").click(function(){
         $(this).prev().find($("select").click());
    });